Miklós Abert | Searcharxiv

arXiv2024

Co-spectral radius for countable equivalence relations

We define the co-spectral radius of inclusions $\mathcal{S}\leq \mathcal{R}$ of discrete, probability measure-preserving equivalence relations, as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. For the proof, we develop a more general version of the 2-3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.

math.PR↗

arXiv2023

Growth dichotomy for unimodular random rooted trees

We show that the growth of a unimodular random rooted tree $(T,o)$ of degree bounded by $d$ always exists, assuming its upper growth passes the critical threshold $\sqrt{d-1}$. This complements Timar's work who showed the possible nonexistence of growth below this threshold. The proof goes as follows. By Benjamini-Lyons-Schramm, we can realize $(T,o)$ as the cluster of the root for some invariant percolation on the $d$-regular tree. Then we show that for such a percolation, the limiting exponent with which the lazy random walk returns to the cluster of its starting point always exists. We develop a new method to get this, that we call the 2-3-method, as the usual pointwise ergodic theorems do not seem to work here. We then define and prove the Cohen-Grigorchuk co-growth formula to the invariant percolation setting. This establishes and expresses the growth of the cluster from the limiting exponent, assuming we are above the critical threshold.

math.PR↗